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Creative Problem Posing in the Middle-Grades Classroom

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Earlier this year, I was introduced to Paul Lockhart’s remarkable text, A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. Lockhart persuasively advocates for mathematics teaching and a mathematics curriculum that provides students with opportunities to ask their own questions and devise their own methods for answering these questions. Taking Lockhart’s message to heart, I asked my eighth-grade daughter, Cassady, to construct a mathematics task that fosters creativity among students. My conversation with Cassady surprised me. Both she and Lockhart have encouraged me to reconsider the nature of mathematics and the way that we share it with students in the middle grades.

Cassady’s Task 

I began by asking Cassady whether mathematics was creative. An edited version of the hourlong conversation is highlighted in the following 10-minute video (http://bit.ly/mathcreative).

Video with 8th Grader
 

Cassady remarked that mathematics in the middle grades is “like a formula, and there’s one correct answer.” In response, she provided me with the following task:

John had 10 sticks. They were 12 cm, 2 cm, 4 cm, 9 cm, 20 cm, 6 cm, 15 cm, 5 cm, 10 cm, and 1 cm long. Can John make a polygon using these 10 sticks? If so, draw it. If not, explain why not.  

Remarkably, the task was not familiar to Cassady. It had not been posed to her in a previous class. The task was truly her own. In fact, she didn’t know if such a polygon could be built. When I asked Cassady if her classmates had opportunities to pose and solve their own problems, Cassady remarked, “We have kind of outgrown it. We don’t do story problems anymore . . . we haven’t done that since third grade.”

Our Solution 

During the next 30 minutes, Cassady and I worked collaboratively on her problem. Unfortunately, we had no hands-on manipulatives, so we attacked the problem using GeoGebra (http://www.geogebra.org/cms/en/), a free dynamic mathematics software program. A version of her sketch can be accessed on GeoGebraTube at http://www.geogebratube.org/student/m112159. 

Cassady began by plotting 8 of the 10 vertices as lattice points on a square grid. She arranged consecutive vertices to form right angles. This made it easier for her to satisfy the length requirements of the task. I’ve labeled these points consecutively as A through I (see fig. 1).

Fig. 1 We approached the solution to this problem using GeoGebra.

 Plot of first 8 vertices  

 (a) Plot of first 8 vertices 

 Constructing a circle to plot 2 remaining vertices  

(b) Constructing a circle to plot 2 remaining vertices 

With 2 lengths remaining (namely, 10 cm and 15 cm), Cassady constructed a circle with radius 10 units centered at the 8th vertex (point I). Point J was constructed on the circle. This step ensured that segment IJ, the 9th side of the polygon, had length 10 units. Next, she constructed JA as the 10th side, dragging J until JA had the desired length of 15 units.

The solution strategy also highlighted Cassady’s strategic use of technology (one of the Common Core’s Standards for Mathematical Practice is Standard 5: Use appropriate tools strategically). GeoGebra played an invaluable role in the problem-solving process. Using a compass and ruler to construct the location of point J would have been time-consuming and error-laden without the software. However, GeoGebra did not trivialize the task. Rather, the problem required a good deal of creative thinking with the software. The idea to use a circle was a “creative breakthrough” that involved considerable mathematics know-how and outside-the-box thinking. In the journey to the solution, Cassady and I discussed a number of interesting concepts, including the triangle inequality and slopes of parallel lines.

Math Is Creative 

At the conclusion of our conversation, Cassady noted that our mathematics work was “definitely” more creative than the activities she does in school. She noted that she “really likes problems where she doesn’t know if there’s a definite answer.” The experience has encouraged me to ask more questions of my students regarding their own learning needs, listening carefully to their answers, then acting on them. As Lockhart notes, “The only people who understand what is going on are the ones most often blamed and least often heard: the students” (p. 11). 


Michael Todd EdwardsMichael Todd Edwards is an associate professor of mathematics education at Miami University in Oxford, Ohio. He is the coeditor of Contemporary Issues in Technology and Mathematics Teacher Education, executive editor of the North American GeoGebra Journal, and codirector of the GeoGebra Institute of Ohio. His research interests focus on the teaching and learning of school mathematics with technology (specifically, dynamic mathematics software), ethical issues surrounding the use of free software and the free software movement, and writing as a vehicle to learn mathematics at all levels of instruction.


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